A Machian wave effect in conformal, scalar-tensor gravitational theory

Woodward proposed that driven mass-energy fluctuations could yield a frequency-dependent “Machian” gravitational response $\propto {\mathrm{\partial }}_t^2{M}_{\mathrm{l}\mathrm{o}\mathrm{c}}(t)$, amplified by a Sciama-scale cosmic potential $\mathrm{\Phi }\mathrm{/}{c}^2\sim -1$. We test this claim covariantly in (i) Einstein gravity and (ii) Hoyle-Narlikar (HN) conformal scalar-tensor gravity. In GR, the Landau-Lifshitz relaxed equations in harmonic gauge contain nonlinear terms of the form $H^{\alpha \beta }{\mathrm{\partial }}_{\alpha }{\mathrm{\partial }}_{\beta }{H}^{\mu \nu }$, including a near-zone piece $H^{00}{\mathrm{\partial }}_t^2{H}^{\mu \nu }$. These terms are not independent matter sources; they arise from expanding the curved wave operator about a flat background. Moving them back to the left-hand side restores the quasilinear principal part, and for laboratory devices their size is suppressed by $\sim (U_N\mathrm{/}{c}^2)(\omega L\mathrm{/}c{)}^2\ll 1$, with no enhancement by any cosmological potential. In HN theory, the conformal scalar satisfies ${\mathrm{\nabla }}_a{\mathrm{\nabla }}^{a}m+(R\mathrm{/}6)m=\lambda N$. For a localized device of size $L$ driven at angular frequency $\omega$, $\mathrm{\mid }(c^{-2}{\mathrm{\partial }}_t^2{m}_s)\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\nabla }}^2{m}_s\mathrm{\mid }\sim (\omega L\mathrm{/}c{)}^2\ll 1$, so the response is effectively instantaneous (Poisson-like), not wave-amplified. Baryon-number conservation fixes the scalar charge, so the rest-mass monopole cannot oscillate; any radiating monopole requires nonconservative internal-energy variations, further suppressed by $E_{\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{/}(M{c}^2)$ and by $M_{\mathrm{d}\mathrm{e}\mathrm{v}}\mathrm{/}{M}_H$. Thus any Mach-effect thrust is far too small for practical propulsion.